How to merge 2 parts of the same object?
Recorded a quick video showing how to easily do it.

The steps are:


Create two loop-cuts on the smallest/thinnest part.
Snap the new loops along the X axis(or whatever axis are appropriate for your mesh) to the mesh you want to merge it with.
Select the faces your mesh is going through and create an inset face.
Scale down the new faces and snap the edges into the thin piece.
When all edges/vertices are overlapping use Alt+M and select "By Distance" to merge overlapping vertices.
Finish off by either manually selecting and deleting the interior faces or use Select - Select all by trait - Interior faces to select them and delete